Pseudo-character of a set

$A$ in a topological space $X$

The smallest infinite cardinal number $\tau$ for which there exists a family of cardinality $\tau$ of sets open in $X$ with intersection $A$. It is usually denoted by $\psi(A,X)$. The pseudo-character $\psi(A,X)$ is defined for all subsets $A$ of $X$ only when all singleton subsets in $X$ are closed. The pseudo-character $\psi(x,X)$ of a point $x\in X$ in a topological space $X$ is understood to be the pseudo-character $\psi(\{x\},X)$ of the set $\{x\}$ in $X$.

The pseudo-character $\psi(X)$ of a topological space $X$ is the smallest infinite cardinal number $\tau$ such that each point is the intersection of a family of cardinality $\leq\tau$ of sets which are open in $X$. Spaces with countable pseudo-character are those in which every point is a $G_\delta$-set (cf. Set of type $F_\sigma$ ($G_\delta$)). Each topological space can be represented as the image under a continuous open mapping of a paracompact Hausdorff space with a countable pseudo-character. For compact Hausdorff spaces the countability of the pseudo-character is equivalent to the first axiom of countability. In general, the pseudo-character of a closed set $A$ in a compact Hausdorff space $X$ is equal to the minimum cardinality of a defining system of neighbourhoods of the set $A$ in $X$.

References

[1]	A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian)
[2]	A.V. Arkhangel'skii, "Classes of topological groups" Russian Math. Surveys , 36 : 3 (1981) pp. 151–174 Uspekhi Mat. Nauk , 36 : 3 (1981) pp. 127–146

References

[a1]	R. Engelking, "General topology" , Heldermann (1989)
[a2]	K. Kunen (ed.) J.E. Vaughan (ed.) , Handbook of set-theoretic topology , North-Holland (1984) pp. Chapts. 1–2